вправа 11.3 гдз 11 клас математика Мерзляк Номіровський 2019
Вправа 11.3
Умова:
Умова:
Обчисліть визначений інтеграл:
Відповідь ГДЗ:
\begin{equation} 1) \int_{5}^{7}xdx=\frac{x^{2}}{2} \mid^{7}_{5} = \end{equation} \begin{equation} =\frac{7^{2}}{2} -- \frac{5^{2}}{2}= \end{equation} \begin{equation} =\frac{49-25}{2}= \end{equation} \begin{equation} =\frac{24}{2}=12; \end{equation} \begin{equation} 2) \int_{5}^{7}dx=x \mid^{8}_{3} 8-3=5; \end{equation} \begin{equation} 3) \int_{-3}^{0}x^{2}dx=\frac{x^{3}}{2} \mid^{0}_{-3}= \end{equation} \begin{equation} =\frac{0^{3}}{3}-\frac{(-3)^{3}}{3}=27=9; \end{equation} \begin{equation} 4)\int_{-1}^{2}x^{4}dx=\frac{x^{5}}{5} \mid^{2}_{-1} = \end{equation} \begin{equation} = \frac{2^{5}}{5}-\frac{(-1)^{5}}{5}= \end{equation} \begin{equation} =\frac{32+1}{5}=\frac{33}{5}; \end{equation} \begin{equation} 5) \int_{0}^{\frac{\pi}{2}}xdx=- \cos x \mid^{\frac{\pi}{3}}_{0}= \end{equation} \begin{equation} =- \cos \frac{\pi}{3}- (- \cos 0)= \end{equation} \begin{equation} =- \frac{1}{2}+1=\frac{1}{2}; \end{equation} \begin{equation} 6) \int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{dx}{\cos^{2} x}= \end{equation} \begin{equation} =tg x \mid^{\frac{\pi}{3}}_{\frac{\pi}{4}}= \end{equation} \begin{equation} =tg\frac{\pi }{3}-tg\frac{\pi }{4}=\sqrt{3}-1; \end{equation} \begin{equation} 7) \int_{16}^{100}\frac{dx}{\sqrt{x}}= \end{equation} \begin{equation} =2\sqrt{x} \mid^{100}_{16} = \end{equation} \begin{equation} =2\sqrt{100}-2\sqrt{16}= \end{equation} \begin{equation} =20-8=12; \end{equation} \begin{equation} 8) \int_{e^{2}}^{e^{3}}\frac{dx}{x}= \ln \left | x \right | \mid^{e^{3}}_{e^{2}}= \end{equation} \begin{equation} =\ln e^{3}=3-2=1; \end{equation} \begin{equation} 9) \int_{1}^{10}\frac{dx}{x^{2}}= \end{equation} \begin{equation} =-\frac{1}{x} \mid^{10}_{1} =- \end{equation} \begin{equation} -\frac{1}{10}-(\frac{1}{1})= \end{equation} \begin{equation} =-\frac{1}{10}+1=\frac{9}{10}; \end{equation} \begin{equation} 10)\int_{-2}^{3} 3^{x}dx=\frac{3^{x}}{\ln 3} \mid^{3}_{-2} = \end{equation} \begin{equation} =\frac{3^{3}}{\ln 3}-\frac{3^{-2}}{\ln 3}= \end{equation} \begin{equation} =\frac{27-\frac{1}{9}}{\ln 3}= \end{equation} \begin{equation} \frac{26\frac{8}{9}}{\ln 3}=\frac{242}{9 \ln 3}. \end{equation}
\begin{equation} 1) \int_{5}^{7}xdx=\frac{x^{2}}{2} \mid^{7}_{5} = \end{equation} \begin{equation} =\frac{7^{2}}{2} -- \frac{5^{2}}{2}= \end{equation} \begin{equation} =\frac{49-25}{2}= \end{equation} \begin{equation} =\frac{24}{2}=12; \end{equation} \begin{equation} 2) \int_{5}^{7}dx=x \mid^{8}_{3} 8-3=5; \end{equation} \begin{equation} 3) \int_{-3}^{0}x^{2}dx=\frac{x^{3}}{2} \mid^{0}_{-3}= \end{equation} \begin{equation} =\frac{0^{3}}{3}-\frac{(-3)^{3}}{3}=27=9; \end{equation} \begin{equation} 4)\int_{-1}^{2}x^{4}dx=\frac{x^{5}}{5} \mid^{2}_{-1} = \end{equation} \begin{equation} = \frac{2^{5}}{5}-\frac{(-1)^{5}}{5}= \end{equation} \begin{equation} =\frac{32+1}{5}=\frac{33}{5}; \end{equation} \begin{equation} 5) \int_{0}^{\frac{\pi}{2}}xdx=- \cos x \mid^{\frac{\pi}{3}}_{0}= \end{equation} \begin{equation} =- \cos \frac{\pi}{3}- (- \cos 0)= \end{equation} \begin{equation} =- \frac{1}{2}+1=\frac{1}{2}; \end{equation} \begin{equation} 6) \int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{dx}{\cos^{2} x}= \end{equation} \begin{equation} =tg x \mid^{\frac{\pi}{3}}_{\frac{\pi}{4}}= \end{equation} \begin{equation} =tg\frac{\pi }{3}-tg\frac{\pi }{4}=\sqrt{3}-1; \end{equation} \begin{equation} 7) \int_{16}^{100}\frac{dx}{\sqrt{x}}= \end{equation} \begin{equation} =2\sqrt{x} \mid^{100}_{16} = \end{equation} \begin{equation} =2\sqrt{100}-2\sqrt{16}= \end{equation} \begin{equation} =20-8=12; \end{equation} \begin{equation} 8) \int_{e^{2}}^{e^{3}}\frac{dx}{x}= \ln \left | x \right | \mid^{e^{3}}_{e^{2}}= \end{equation} \begin{equation} =\ln e^{3}=3-2=1; \end{equation} \begin{equation} 9) \int_{1}^{10}\frac{dx}{x^{2}}= \end{equation} \begin{equation} =-\frac{1}{x} \mid^{10}_{1} =- \end{equation} \begin{equation} -\frac{1}{10}-(\frac{1}{1})= \end{equation} \begin{equation} =-\frac{1}{10}+1=\frac{9}{10}; \end{equation} \begin{equation} 10)\int_{-2}^{3} 3^{x}dx=\frac{3^{x}}{\ln 3} \mid^{3}_{-2} = \end{equation} \begin{equation} =\frac{3^{3}}{\ln 3}-\frac{3^{-2}}{\ln 3}= \end{equation} \begin{equation} =\frac{27-\frac{1}{9}}{\ln 3}= \end{equation} \begin{equation} \frac{26\frac{8}{9}}{\ln 3}=\frac{242}{9 \ln 3}. \end{equation}